The problem asks us to calculate the volume integral of the divergence of a given vector field \(\vec E\) over a unit cube. The vector field is defined as \(\rm \vec E (x, y, z) = 2x^2 \hat i + 5y \hat j + 3 z \hat k\), and the unit cube is defined by the limits \(0 \le x \le 1\), \(0 \le y \le 1\), and \(0 \le z \le 1\). We need to evaluate the integral \(\iiint_V (\vec \nabla \cdot \vec E) dV\).

Calculate Vector Field Divergence

First, we need to find the divergence of the vector field \(\vec E\). The divergence of a vector field \(\vec E = E_x \hat i + E_y \hat j + E_z \hat k\) is given by the formula:

$$ \vec \nabla \cdot \vec E = \frac{\partial E_x}{\partial x} + \frac{\partial E_y}{\partial y} + \frac{\partial E_z}{\partial z} $$

For the given vector field \(\rm \vec E (x, y, z) = 2x^2 \hat i + 5y \hat j + 3 z \hat k\), we have:

• \( E_x = 2x^2 \)
• \( E_y = 5y \)
• \( E_z = 3z \)

Now, we calculate the partial derivatives:

• \( \frac{\partial E_x}{\partial x} = \frac{\partial}{\partial x}(2x^2) = 4x \)
• \( \frac{\partial E_y}{\partial y} = \frac{\partial}{\partial y}(5y) = 5 \)
• \( \frac{\partial E_z}{\partial z} = \frac{\partial}{\partial z}(3z) = 3 \)

Adding these together, the divergence is:

$$ \vec \nabla \cdot \vec E = 4x + 5 + 3 = 4x + 8 $$

Evaluate the Volume Integral

Now we need to evaluate the volume integral of this divergence over the unit cube V:

$$ \iiint_V (\vec \nabla \cdot \vec E) dV = \iiint_V (4x + 8) dV $$

The limits for the unit cube are \(0 \le x \le 1\), \(0 \le y \le 1\), and \(0 \le z \le 1\). So the integral becomes a triple integral:

$$ \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} (4x + 8) \, dx \, dy \, dz $$

Step-by-Step Integration

We perform the integration step-by-step:

1. Integrate with respect to \(x\): $$ \int_{0}^{1} (4x + 8) \, dx = \left[ \frac{4x^2}{2} + 8x \right]_{0}^{1} = \left[ 2x^2 + 8x \right]_{0}^{1} $$ $$ = (2(1)^2 + 8(1)) - (2(0)^2 + 8(0)) = (2 + 8) - 0 = 10 $$
2. Integrate the result with respect to \(y\): The result from the first step is 10. Now integrate this constant value with respect to \(y\): $$ \int_{0}^{1} 10 \, dy = \left[ 10y \right]_{0}^{1} = 10(1) - 10(0) = 10 $$
3. Integrate the result with respect to \(z\): The result from the second step is 10. Now integrate this constant value with respect to \(z\): $$ \int_{0}^{1} 10 \, dz = \left[ 10z \right]_{0}^{1} = 10(1) - 10(0) = 10 $$

Thus, the value of the volume integral \(\iiint_V (\vec \nabla \cdot \vec E) dV\) is 10.